Aspherical Space

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9/25/2006
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In topology, a branch of mathematics, an aspherical space is a topological space with all homotopy groups π n ( X ) {\displaystyle \pi _{n}(X)} equal to 0 when n > 1 {\displaystyle n>1} . If one works with CW complexes, one can reformulate this condition: an aspherical CW complex is a CW complex whose universal cover is contractible. Indeed, contractibility of a universal cover is the same, by Whitehead's theorem, as asphericality of it. And it is an application of the exact sequence of a fibration that higher homotopy groups of a space and its universal cover are same. (By the same argument, if E is a path-connected space and p : E → B {\displaystyle p\colon E\to B} is any covering map, then E is aspherical if and only if B is aspherical.) Each aspherical space X is, by definition, an Eilenberg–MacLane space of type K ( G , 1 ) {\displaystyle K(G,1)} , where G = π 1 ( X ) {\displaystyle G=\pi _{1}(X)} is the fundamental group of X. Also directly from the definition, an aspherical space is a classifying space for its fundamental group (considered to be a topological group when endowed with the discrete topology).
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Aspherical manifold
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